OFFSET
1,2
COMMENTS
log(A003418(n)) = Sum_{k>=1} (T(n, k)/k - 1/k).
Partial sums of the symmetric matrix A191898. - Mats Granvik, Apr 12 2020
1 + Sum_{k=1..2*n} sign((sign(n+Sum_{j=2..k}-|T(n,j)|)+1)) appears to be asymptotic to sqrt(8*n). - Mats Granvik, Jun 08 2020
From Mats Granvik, Apr 14 2021: (Start)
Conjecture 1: For n>1: max(T(1..n,n)) + min(T(1..n,n)) = 2*mean(T(1..n,n)) = -A023900(n).
Patterns that eventually fail or possibly become switched are:
max(T(n,1..n!)) = 1,2,3,4,5,6,7,8,...
min(T(n,1..n!)) = 1,0,-2,-3,-7,-5,-11,-12,...
which are the first 8 terms of A275205.
Conjecture 2: The Prime Number Theorem should imply: mean(T(n,1..n!)) = 1.
(End)
LINKS
Mats Granvik, Attempt at proof of the conjectured square root order asymptotics for the sequence constructed from this matrix.
Mats Granvik, Mathematica MatrixPlot of 1000 times 1000 size matrix
Mats Granvik, Mathematica program for the recurrence
Mathematics Stack Exchange, Do these series converge to the von Mangoldt function?
FORMULA
Recurrence:
T(n, 1) = [n >= 1]*n;
T(1, k) = 1;
T(n, k) = [n > k]*T(n - k, k) + [n <= k](Sum_{i=0..n-1} T(n - 1, k - i) - Sum_{i=1..n-1} T(n, k - i)). - Mats Granvik, Jun 19 2020
T(n,k) = Sum_{i=1..n} A191898(i,k).
EXAMPLE
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...
3, 1, 0, 1, 3, -2, 3, 1, 0, 1, 3, -2, 3, 1, ...
4, 0, 1, 0, 4, -3, 4, 0, 1, 0, 4, -3, 4, 0, ...
5, 1, 2, 1, 0, -2, 5, 1, 2, -4, 5, -2, 5, 1, ...
6, 0, 0, 0, 1, 0, 6, 0, 0, -5, 6, 0, 6, 0, ...
7, 1, 1, 1, 2, 1, 0, 1, 1, -4, 7, 1, 7, -6, ...
8, 0, 2, 0, 3, 0, 1, 0, 2, -5, 8, 0, 8, -7, ...
9, 1, 0, 1, 4, -2, 2, 1, 0, -4, 9, -2, 9, -6, ...
10, 0, 1, 0, 0, -3, 3, 0, 1, 0, 10, -3, 10, -7, ...
11, 1, 2, 1, 1, -2, 4, 1, 2, 1, 0, -2, 11, -6, ...
12, 0, 0, 0, 2, 0, 5, 0, 0, 0, 1, 0, 12, -7, ...
13, 1, 1, 1, 3, 1, 6, 1, 1, 1, 2, 1, 0, -6, ...
14, 0, 2, 0, 4, 0, 0, 0, 2, 0, 3, 0, 1, 0, ...
...
MATHEMATICA
f[n_] := DivisorSum[n, MoebiusMu[#] # &]; nn = 14; A = Accumulate[Table[Table[f[GCD[n, k]], {k, 1, nn}], {n, 1, nn}]]; Flatten[Table[Table[A[[n - k + 1, k]], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Jun 09 2020 *)
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Mats Granvik, Aug 10 2019
STATUS
approved